Question

Choose the correct number of ways in which $15$different books can be divided into five heaps of an equal number of books

• A. $\frac{15!}{\left[5!{\left(3!\right)}^5\right]}$
• B. $\frac{15!}{{\left(3!\right)}^5}$
• C. ${}^{15}C_5$
• D. ${}^{15}P_5$

Answer 1

The correct option is A

$\frac{15!}{\left[5!{\left(3!\right)}^5\right]}$

Explanation for the correct option:

To find the total number of ways:

There are five heaps of an equal number of books.

Thus each heap contains $\frac{15}{5}=3$books.

So the required number of ways

$$\begin{gathered} ={}^{15}C_3\times {}^{12}C_3\times {}^{9}C_3\times {}^{6}C_3\times {}^{3}C_3\times \frac{1}{5!} \\ =\frac{15!}{12!\times 3!}\times \frac{12!}{9!\times 3!}\times \frac{9!}{6!\times 3!}\times \frac{6!}{3!\times 3!}\times \frac{3!}{0!\times 3!}\times \frac{1}{5!} \\ =\frac{5!}{[5!{(3!)}^5]} \end{gathered}$$

Hence ,the correct option is (A).